Probing deformed commutators with
macroscopic harmonic oscillators
Francesco Marin
HUMOR
Heisenberg Uncertainty Measured with Opto-mechanical Resonators
Morning Session (2): Strong field tests of General Relativity (Pulsars, Black holes,...)
Phenomenological quantum gravity

General ‘remark’: one cannot determine a position with an accuracy better
than the Planck length LP = hG/c3 = 1.6 10-35 m
 Generalized Heisenberg uncertainty relations (GUP)
 Generalized commutators between p e q
 Modified quantum physics
Detecting signatures of Planck scale-physics in
highly-sensitive metrological systems
Basic assumptions:
Heisenberg dynamics
Deformed commutation relations
from
Solution:
3° harmonic
Freq. shift
 Test on a wide mass range
 High mechanical quality factor  ‘isolated’ oscillators
 Exploit the slow decay to obtain frequency/3° harmonic
vs amplitude curves
1° oscillator:
m1g
2° oscillator:
m  100 g
3° oscillator:
m  100 ng
SiN membrane
0.5 x 0.5 mm2 x 50nm
mass = 135 ng
Q = 8.6x105
m = 20 μg
fm = 141 KHz
Q = 1.2 x106
T = 4.3 K
Evidence of deformed commutator?
Structural non-linearity
‘Model-independent’ limits
MP
H atom
MP
H atom
AURIGA
MP
H atom
AURIGA
Equivalence principle
MP
H atom
AURIGA
Equivalence principle
3° harmonic
dw vs q
M. Bavaj. et al., arXiv: 1411.6410
What’s the meaning of our measurements?
(my poor man view)
We have a Planck energy concentrated on a Planck length?
NO
… but we are are searching tiny ‘residual’ effects
Which models are we limiting/questioning?
Strictly speaking, NO ONE : no model predicts a deformed
commutator for macroscopic variables
… but is there any satisfactory model?
If a deformed commutator applies to the coordinates of a fundamental
constituent, then its effect on a macroscopic object (composed of N
such constituents) should decrease as 1/N
• what is a fundamental constituent?
• geometrical properties of space-time  property of
each particle
… A different point of view: a foamy space-time
If a deformed commutator applies to the coordinates of a fundamental
constituent, then its effect on a macroscopic object (composed of N
such constituents) should decrease as 1/N
• what is a fundamental constituent?
• geometrical properties of space-time  property of each particle
• not in the spirit of quantum mechanics: the position and momentum of
an oscillator c.m. should be THE meaningful (i.e., measurable)
quantities. Uncertainty relation between p and q is tightly related to the
general problem of quantum measurements (see Braginsky, Caves, etc.)
Focus on peculiar quantum properties
Tracks in the quantum+GR field are diversified …
… we are just setting some experimental poles